Question

the following situation involves a rate of change that is constant. write a statement that describes how one variable changes with respect to the other, give the rate of change numerically (with units), and use the rate of change rule to answer any questions.
a 1 - degree change (increase or decrease) on the celsius temperature scale is equivalent to a $\frac{9}{5}$ degree change on the fahrenheit temperature scale. how much does the fahrenheit temperature increase if the celsius temperature increases 9 degrees? how much does the fahrenheit temperature decrease if the celsius temperature decreases 20 degrees?
which statement describes this situation?
a. the celsius temperature varies with respect to the fahrenheit temperature with a rate of change of $\frac{9}{5}^{circ}$f per degree c.
b. the fahrenheit temperature varies with respect to the celsius temperature with a rate of change of $\frac{9}{5}^{circ}$c per degree f.
c. the celsius temperature varies with respect to the fahrenheit temperature with a rate of change of $\frac{9}{5}^{circ}$c per degree f.
d. the fahrenheit temperature varies with respect to the celsius temperature with a rate of change of $\frac{9}{5}^{circ}$f per degree c.

Explanation:

Step1: Identify the rate - change relationship

We know that a 1 - degree change in Celsius is equivalent to a $\frac{9}{5}$ - degree change in Fahrenheit. This means the Fahrenheit temperature changes with respect to the Celsius temperature. The rate of change of Fahrenheit temperature ($F$) with respect to Celsius temperature ($C$) is $\frac{9}{5}^{\circ}F$ per degree $C$.

Step2: Analyze the multiple - choice options

Option A has the variables in the wrong order (it should be $F$ with respect to $C$, not $C$ with respect to $F$). Option B has the wrong units for the rate of change (it says $\frac{9}{5}^{\circ}C$ per degree $F$ instead of $\frac{9}{5}^{\circ}F$ per degree $C$). Option C has the wrong order of variables and wrong units. Option D is correct as it states that the Fahrenheit temperature varies with respect to the Celsius temperature with a rate of change of $\frac{9}{5}^{\circ}F$ per degree $C$.

Step3: Calculate the change in Fahrenheit for given Celsius changes

If the Celsius temperature increases by $\Delta C = 9$ degrees, using the rate of change formula $\Delta F=rate\times\Delta C$. Substituting the rate $\frac{9}{5}^{\circ}F$ per degree $C$ and $\Delta C = 9$ degrees, we get $\Delta F=\frac{9}{5}\times9=\frac{81}{5}=16.2$ degrees Fahrenheit.
If the Celsius temperature decreases by $\Delta C = 20$ degrees, then $\Delta F=\frac{9}{5}\times(- 20)=-36$ degrees Fahrenheit (the negative sign indicates a decrease).

Answer:

D. The Fahrenheit temperature varies with respect to the Celsius temperature with a rate of change of $\frac{9}{5}^{\circ}F$ per degree $C$.
The Fahrenheit temperature increases by $16.2$ degrees if the Celsius temperature increases 9 degrees.
The Fahrenheit temperature decreases by 36 degrees if the Celsius temperature decreases 20 degrees.